Problem 81 - Intermediate Algebra

Question

Rationalize each denominator. All variables represent positive real numbers. $$ \sqrt[3]{\frac{4}{81}} $$

Step-by-Step Solution

Answer

The rationalized expression is \( \frac{\sqrt[3]{36}}{3} \).

Step 1: Identify the Expression

We have the expression \( \sqrt[3]{\frac{4}{81}} \). Here, the goal is to rationalize the denominator such that there are no cube roots.

Step 2: Separate the Fractional Cube Root

Express the cube root of the fraction as the quotient of cube roots: \( \frac{\sqrt[3]{4}}{\sqrt[3]{81}} \).

Step 3: Simplify the Denominator

Recognize that \( 81 = 3^4 \) and therefore, \( \sqrt[3]{81} = (3^4)^{\frac{1}{3}} = 3^{\frac{4}{3}} = 3 \cdot \sqrt[3]{3} \).

Step 4: Rationalize the Denominator

Multiply both the numerator and the denominator by \( \sqrt[3]{9} = (3^2)^{\frac{1}{3}} \) to eliminate the cube root in the denominator. This gives:\[ \frac{\sqrt[3]{4} \cdot \sqrt[3]{9}}{\sqrt[3]{81} \cdot \sqrt[3]{9}} = \frac{\sqrt[3]{36}}{3} \].

Step 5: Write the Simplified Expression

The rationalized expression is \( \frac{\sqrt[3]{36}}{3} \), where \( 3 \) is free of any cube roots.

Key Concepts

Cube Roots

When we talk about cube roots, we're exploring how to find a number that, when multiplied by itself twice, gives the original number. For example, the cube root of 8 is 2 because 2 multiplied by itself three times (2 × 2 × 2) equals 8.
Cube roots are represented using the radical symbol with a small 3, like this: \( \sqrt[3]{x} \). In our exercise, we deal with the cube root \( \sqrt[3]{\frac{4}{81}} \), which asks us to determine the cube root of the entire fraction.
To handle a fractional cube root, each part of the fraction can be treated separately, giving us \( \frac{\sqrt[3]{4}}{\sqrt[3]{81}} \).
This step compartmentalizes the cube root problem, allowing us to address the numerator and denominator individually, making it easier to conjugate the cube root.

Fractional Exponents

Fractional exponents are another way to express roots. When you see an exponent like \( x^{m/n} \), it means taking the \( n^{th} \) root of \( x \), and then raising the result to the \( m^{th} \) power.
For instance, \( x^{1/3} \) indicates the cube root of \( x \): \( \sqrt[3]{x} \).
So when simplifying our expression \( \sqrt[3]{81} \), we convert it to its fractional exponent form \( (81)^{1/3} \). Recognizing that 81 is \( 3^4 \), we can write \( 81^{1/3} \) as \( (3^4)^{1/3} = 3^{4/3} = 3 \times 3^{1/3} \).
This step illustrates the value of understanding fractional exponents, particularly in breaking down complex roots into more manageable parts.

Simplifying Expressions

Once our fraction is broken into parts, the focus shifts to simplifying the expression to make it easier to work with. Our goal with rationalizing denominators, as outlined in the exercise, is to eliminate roots from the denominator entirely.
After separating \( \frac{\sqrt[3]{4}}{\sqrt[3]{81}} \), and knowing that \( \sqrt[3]{81} = 3 \times \sqrt[3]{3} \), we need to multiply both the numerator and denominator by \( \sqrt[3]{9} \) — which corresponds to \( (3^2)^{1/3} \).
This allows us to cancel out the cube root of the denominator. The simplification yields \( \frac{\sqrt[3]{36}}{3} \). The process simplifies further, if needed, to ensure the denominator is a simple whole number, devoid of cubic roots, culminating in a more refined and practical expression for use in further calculations.